I am looking for a reference for the following question:

To let $ mathfrak {g} $ be a split semisimple couch over algebra $ mathbb {R} $ with corresponding split semisimple adjoint group $ G $, Write $ B = mathfrak {g} : // , G $ for the adjoint quotient a variety is isomorphic too $ mathbb {A} ^ { text {rank} (G)} $ parameterization $ G $invariant polynomials of $ mathfrak {g} $, Write $ B ^ {rs} subset B $ for the open subscheme of the regular semisimple elements the complement of the discriminant polynomial.

To let $ mathfrak {h} subset mathfrak {g} $ be a Cartan subalgebra that does not necessarily have to be shared, and write $ mathfrak {h} ^ {rs} subset mathfrak {h} $ for the regular semisimple locus of $ mathfrak {h} $,

Since the morphism $ mathfrak {h} ^ {rs} rightarrow B ^ {rs} $ is finally, the corresponding morphism $ pi: mathfrak {h} ^ {rs} ( mathbb {R}) rightarrow B ^ {rs} ( mathbb {R}) $ at real points (with the Euclidean topology) there will be a local homeomorphism with finite fibers. If $ U subset B ^ {rs} ( mathbb {R}) $ is then either a connected component $ pi ^ {- 1} (U) $ is empty of $ pi ^ {- 1} (U) rightarrow U $ is a cover room.

Question:is $ pi ^ {- 1} (U) rightarrow U $ a trivial cover space?

That's true, though $ mathfrak {h} $ is split because in this case the Weyl group acts transitive on the connected components of $ mathfrak {h} ^ {rs} ( mathbb {R}) $, Another way of formulating my question is whether it is true that the true points of the Weyl group of $ mathfrak {h} $ acts transitive on the connected components of $ mathfrak {h} ^ {rs} ( mathbb {R}) $,

One solution would be to show that every connected component of $ B ^ {rs} ( mathbb {R}) $ is capable of contracting, but could not prove it. Every help is appreciated!